Eigenvalue Finite Difference Approximations for Regular and Singular Sturm-Liouville Problems

نویسندگان

  • Nabil R. Nassif
  • NABIL R. NASSIF
چکیده

This paper includes two parts. In the first part, general error estimates for "stable" eigenvalue approximations are obtained. These are practical in the sense that they are based on the discretization error of the difference formula over the eigenspace associated with the isolated eigenvalue under consideration. Verification of these general estimates are carried out on two difference schemes: that of Numerov to solve the Schrödinger singular equation and that of the central difference formula for regular Sturm-Liouville problems. In the second part, a sufficient condition for obtaining a "stable" difference scheme is derived. Such a condition (condition (N) of Theorem 2.1) leads to a simple "by hand" verification, when one selects a difference scheme to compute eigenvalues of a differential operator. This condition is checked for oneand two-dimensional problems. Introduction. In this work, we are concerned with eigenvalue-eigenvector approximation by finite difference methods for differential operators defined on functions with bounded or unbounded domains. Our results will be illustrated in particular for the Schrödinger radial operator whose "energy levels" are obtained numerically using difference schemes. Let (1.1) L[y] = -y" + q(x)y, 0 < x < oo, and consider the boundary conditions (1.2) B[y} = cy'(0) + dy(0) = 0 and (1.3) y(x) bounded on (0, oo). Let x, = ih, 0 < /' < 7Y, x0 = 0, X = xN = Nh, withlimA_0 X = lim^o TV = oo. Optimal error estimates for difference methods will depend on how X(h) and N(h) tend to oo. For example (see Corollary 2.1), possible choices for X(h) and N(h) are, respectively, m2 and 2mm2, with h = l/2m. The Numerov [8] difference scheme consists in finding Y = {Y¡}0<¡<N, Xh e R, such that (-y,., + 27, Yi+1)/h2 +(9,_1yi_1 + 10,7,7, + qi+1Yi+l)/12 = Xh(Yi_l + lOY,+ Yi+l)/\2, (1.5) Bh[Y] = 0, Received November 4, 1982; revised January 16, 1984 and October 8, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 65L15: Seondary 34B25. ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page 561 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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تاریخ انتشار 2010